# Macaulay's Duration

Macaulay’s duration is a measure of a bond price sensitivity to changes in market interest rates. It is calculated as the weighted-average of the time difference of the bond cash flows from time 0. A high duration means the bond has a high interest rate risk and vice versa.

Macaulay’s duration is the most basic measure of duration. The modified duration and effective duration are a better measures of interest rate risk.

The interest rate risk is a function of how farther the cash flows of a bond are from zero. A zero-coupon bond has higher interest rate risk that a coupon-paying bond of the same maturity. The Macaulay’s duration assesses the interest rate risk with reference to the duration of a zero-coupon bond. The duration of a zero-coupon bond equals its time duration, i.e. its maturity (in years). The duration of a coupon-paying bond can be calculated by considering each cash flow as a separate zero-coupon bond.

## Formula

The maturity (in years) of each cash flow of a coupon bond is weighted based on the proportion of the present value of the cash flow to the total present value of all cash flows. This can be expressed mathematically as follows:

$$\text{Macaulay Duration}\\=\frac{{\text{PV}} _ \text{1}}{\text{PV}}\times \text{T} _ \text{1}+\frac{{\text{PV}} _ \text{2}}{\text{PV}}\times \text{T} _ \text{2}+\text{...}+\frac{{\text{PV}} _ \text{n}}{\text{PV}}\times \text{T} _ \text{n}$$

Where PV1, PV2 and PVn refer to the present value of cash flows that occur T1, T2 and Tn years in future and PV is the price of the bond i.e. the sum of present value of all the bond cash flows at time 0.

The Macaulay’s duration can also be calculated using Microsoft Excel MDURATION function.

## Example

You work as an analyst at a pension fund. The portfolio manager that you report to wants to execute a duration-matching strategy by matching duration of the fund assets with fund liabilities. He has asked you to calculate Macaulay’s duration of the following bonds and identify which bond has the higher interest rate risk:

• Bond A: $1,000 face value coupon bond with 4 and half years till maturity. • Bond B: 5-year$1,000 face value bond paying 5% annual coupon yielding 5.2%

Duration of Bond A is 4.5, i.e. the maturity period (in years) of the zero-coupon bond.

Duration of Bond B is calculated by first finding the present value of each of the annual coupons and maturity value. Annual coupon is $50 (i.e. 5% of the$1,000) and the maturity value is $1,000. The present values of each coupon and its proportion to the total present value of the bond are worked out as follows: Coupon No. 1 2 3 4 5 Total Years till the coupon payment 1 2 3 4 5 Coupon amount$50.00 $50.00$50.00 $50.00$50.00
Maturity value $-$- $-$- $1,000.00 Total cash flow$50.00 $50.00$50.00 $50.00$1,050.00
Present value @ 5.2% $47.53$45.18 $42.95$40.82 $814.91$991.39
Proportion (i.e PV/Price) 4.79% 4.56% 4.33% 4.12% 82.20%

Duration of Bond B equals the years till each cash flow weighted based on percentages calculation above:

$$\text{D}=\text{4.79%}\times\text{1}+\text{4.5%}\times\text{2}+\text{4.33%}\times\text{3}+\text{4.12%}\times\text{4}+\text{82.20%}\times\text{5}=\text{4.54}$$

Because Bond B has slightly higher duration, it has higher interest rate risk. Further, because its cash flows are more dispersed, it is relatively difficult to create a low-maintenance hedge with such a bond.